Kenneth Arrow, co-recipient of the 1972 Nobel Memorial Prize in Economic Sciences, passed away in February. This column outlines the ideas of one of the transcendent minds in the history of economics. The author, holder of a chair named in Arrow’s honour, notes that while his contributions were central in creating much of what constitutes modern quantitative social science, he was always profoundly aware of the limitations of the edifice, constantly seeking to challenge and broaden economic theory.

The recent death of Kenneth Arrow (who was born on 23 August 1921) represents both the loss of one of the transcendent minds in the history of economics, and the closing of a golden age of economic theory. That age – which includes such historically important figures as Arrow’s fellow Nobel laureates Paul Samuelson and Gary Becker – represented a development and expansion of formal economic theory that brought unprecedented precision to the logical foundations of social science.

The economic approach to individual decision-making is derived from the interplay of preferences, constraints, and beliefs. This approach, when combined with the conceptualisation of observed outcomes for an economic environment as equilibria, allows for clear understanding of how markets create and adjudicate interdependences in these decisions. Arrow and the larger body of scholars in this golden age both developed the logical foundations of ideas whose origins go back to Adam Smith and the beginnings of economics, and extended the domain of economics to contexts far beyond markets of conventional supply and demand.

Arrow’s contributions span virtually all of economic theory, but they can be approximated as falling into five distinct areas.

The impossibility theorem

Arrow’s most famous scholarly achievement is his celebrated ‘impossibility theorem’, which lies at the heart of understanding how a government, or other collective decision-making process, can employ individual preferences as inputs from which decisions are determined. Dictatorships can do this with ease: a leader’s preferences determine action. Other cases are far more complex.

Intuitively, democracies aspire to assign equal weight to voters and produce policies that are the ‘will of the people’. But how should differences in preferred outcomes be evaluated? Should everyone have one vote per election, regardless of the intensity of their feelings about the issues at stake? Should voting be first past the post, or should proportional representation be followed? There are many ways to produce collective choices.

We have powerful intuitions as to how collective choice procedures should function. One is that if everyone prefers one policy to another, the latter should never be preferred. Another desideratum is that voting procedures should always produce coherent decisions. But as the Marquis de Condorcet showed in 1785, incoherence can occur in a majority voting system where choices are sequentially considered pairwise: specifically, that it is possible that candidate A defeats B, B defeats C, but C defeats A, thereby failing to produce a coherent notion of a winner.

Condorcet’s remarkable insight leads to the question of whether alternative voting schemes can avoid such outcomes. Arrow’s even more remarkable (1951a) analysis, which was his doctoral dissertation, asked whether there can be any procedure that respects the preferences of all and at the same time always produces coherent decisions? The impossibility theorem proved that the answer is no. Any procedure, no matter how clever, runs the risk of producing cycles in voting or other bizarre outcomes. In other words, it may not be the case that there is always a coherent voice of the people.

Why is this so important? Arrow’s results in no way call into question the intrinsic value of democratic processes. Rather, they demonstrate that no procedure can aggregate individual preferences in a way that meets all objectives in all cases. A perfect voting system or collective action scheme, in this sense, does not exist and any institutional design must recognise this.

General equilibrium theory

Arrow’s second great set of contributions revolves around general equilibrium theory, the high altar of mathematical economics. The economic theory of market economies is based on the idea that individual buyers and sellers face incentives to consume more or less, based on the prices of goods and services. Prices, in turn, adjust in order to balance supply and demand in individual markets.

But what happens in the economy as a whole? Does there exist a set of prices that equates supply and demand across markets? This is the question of whether there is an equilibrium for the whole economy, and if an equilibrium exists, is it socially desirable?

Adam Smith’s ‘invisible hand’ argues that the decentralised decisions of buyers and sellers create collective economic outcomes that are desirable. Is this logically possible? In the 1880s, the great economist Leon Walras recognised that the mathematical representation of individual choices and markets interactions would allow such questions to be answered. Walras set an agenda for mathematical economics whose resolution waited until Arrow and the 1950s.

Together with his sometime co-author and fellow Nobel laureate Gerard Debreu, as well as Lionel McKenzie, Arrow identified conditions for the nature of individual consumers and producers such that an economy can be in general equilibrium; in other words, it is logically possible for supply to equal demand in every market. Arrow and Debreu (1954) gave an affirmative answer to the existence question by describing when equilibrium is possible.

Further, Arrow (1951b) provided extensions and proofs of the celebrated welfare theorems of economics, which show that the equilibrium levels of supply and demand are optimal in the sense that it is not possible for everyone to be made better off by different levels of production and consumption, and that certain types of transfers can move an economy from one efficient outcome to another. These theorems make the invisible hand idea precise and again show when Smith’s profound insights can hold.

But to show that something is logically possible does not mean that it holds in reality. To prove that an equilibrium exists and that it is efficient is to say that under a set of assumptions about how individuals make choices and how markets work, these things are true. Arrow’s monumental achievement was to reveal the logic underlying ideas about market equilibrium and the invisible hand.

Understanding the conditions under which a market economy is efficient also reveals when it is not. By implication, this understanding also reveals how deviations from certain idealisations of how markets function determine the extent to which equilibria are not socially optimal.

As such, Arrow’s work in economic theory moves beyond tired dichotomies of whether markets are good or bad to understanding what they collectively can do. It is really no surprise that Arrow had a longstanding sympathy with socialism in his early years and remained an ardent liberal all his life. His 1978 essay on socialism is fascinating in illustrating how profoundly humane ideals interacted with economic science to produce Arrow’s political commitments.

Decision-making under uncertainty

Arrow’s third great achievement was systematic exploration of the effects of uncertainty on economic decision-making and aggregate outcomes. This work took two stages. First, Arrow developed much of the theory of decision-making under uncertainty and the implications of uncertainty for understanding market equilibria. Part of this involved determining how a rational agent would employ information in an uncertain world to make choices.

Other work characterised the nature of risk at both the individual level and the market level. At the individual level, where his thinking is well surveyed in Arrow (1971), he is most famous for developing (1965), simultaneously with John W. Pratt, the Arrow-Pratt measures of risk aversion, which provide precise characterisation of how uncertainty in consumption affects an individual’s utility. These measures are conventionally used in empirical research.

In terms of market equilibria, Arrow (1964) developed the idea that financial assets can be thought of as ‘contingent commodities’ – that is, objects whose values are contingent on the resolution of uncertainty. From this perspective, one can understand the diversification of risk as the purchase of a set of contingent commodities that together yield the same return regardless of how uncertainty unfolds.

Commodities that only pay off for one of the possible ways that uncertainty unfolds, but not otherwise, are now called Arrow securities. Arrow securities are essential to the modern theory of finance because actual financial assets can be reinterpreted as representing combinations of Arrow securities. Hence the prices of Arrow securities may be used to determine how all assets are priced. This formalisation allowed the Arrow-Debreu model of general equilibrium to generalise naturally to account for uncertainty. These ideas underpin the modern theory of finance.

The extension of general equilibrium theory to uncertainty is a perfect example of Arrow as scientist: rigorous identification of limitations to economic theory combined with constructive solutions via mathematical formalism.

Imperfect information

Arrow’s later work focused on the implications of uncertainty and imperfect information. The ideas here are so rich as to define a fourth area of his contributions.

One of his most celebrated papers is “Uncertainty and the Welfare Economics of Medical Care” (1963). This study identified why markets for medical care are virtually certain to fail to fulfil the conditions under which a market equilibrium is socially desirable. Relative to current debates, Arrow recognised the problem of moral hazard in the behaviour of doctors and patients. All contemporary scholarly analyses of health insurance have been influenced by Arrow’s arguments, in this case none of which were made using mathematics.

Arrow’s focus on the implications of imperfect information led to him to help pioneer (along with Edmund Phelps, another Nobel laureate) the theory of statistical discrimination (1973). Statistical discrimination asks how the socioeconomic outcomes of a group are affected by stereotypes. In a pool of job applicants, suppose an employer cannot observe individual ability but can observe ethnicity. Then an employer who is attempting to forecast performance can be said rationally to ascribe the average performance of individuals of an ethnic group to each member.

If one group starts with higher average educational quality, this will mean its members will receive job offers more often. In turn, this distorts the incentives of individuals to pursue higher quality education and can create a vicious cycle as disadvantaged groups rationally do not make educational investments.

Statistical discrimination is the leading competitor to models of taste-based discrimination, in which African Americans, for example, might be disadvantaged by animus. The model represented an enormous intellectual leap as it identified how discrimination can occur even when bigotry is absent.

Economics of knowledge

Finally, Arrow made two fundamental contributions to the economics of knowledge. First, he provided a clear economic justification for government sponsorship of science. Arrow’s (1962a) argument derives from the public good nature of advances in knowledge. The benefits of such advances cannot be fully captured by the developer of an idea, and so science is a clear example of a case where the conditions under which efficiency of market equilibrium do not hold.

Second, Arrow (1962b) developed a formal theory of how knowledge grows because of economic activity. In his famous model of the economics of learning-by-doing, he argued that the growth of technical knowledge should not be understood as the product of random insights and inspirations by scientists and others, but as a consequence of the environment produced by economic activity. This means that economic growth can beget growth. Arrow’s vision preceded by two decades the emergence of modern endogenous growth theory, pioneered by Paul Romer and Nobel laureate Robert Lucas.

The prescience of Arrow’s work on knowledge is but one example of his living legacy. Alfred North Whitehead once said that European philosophy was a series of footnotes to Plato. Saying the same of Arrow and economics would be an injustice to the achievements of modern researchers (and one that would have mortified Arrow). What is true is that his body of writings has proven to be visionary in many, many directions – so that the most profound research of today has antecedents in Arrovian thought.

Challenging and broadening economic theory

In the latter part of his career, Arrow was deeply involved in the Santa Fe Institute, the leading centre for complexity research. He also (with me) directed the John D. MacArthur Research Network on Social Interactions and Economic Outcomes. He was a regular participant in debates about the environment and climate change and a long-time fellow of the Beijer Institute for Ecological Economics.

What links these activities? In each case, the research endeavour involved challenges to the assumptions and methods of the same neoclassical economic theory that Arrow had constructed.

This is perhaps the ultimate testament to Arrow’s genius. Having created so much of what constitutes modern quantitative social science, he was always profoundly aware of the limitations of the edifice. With no sacrifice of the logical rigour that places his contributions in the realm of permanent changes in knowledge, Arrow never ceased critically evaluating and challenging economic theory. His remarkable 1974 book, The Limits of Organization, exemplifies the Arrow vision of an eventual social science in which disciplinary boundaries between economics, sociology, and psychology dissolve in the quest to achieve a better match with complex reality.

Arrow’s astonishing capacity for critical engagement with economics lives on in the four living Nobel laureates whom he advised. Eric Maskin, Roger Myerson, Alvin Roth, and Michael Spence have each changed economic theory because of their challenges to standard assumptions and their profound insights follow the Arrow example of using impeccable formal logic as the mainspring for progress in social science.

The fifth Nobel laureate he supervised, John Harsanyi, was one of the fathers of game theory, which represents a distinct edifice of microeconomics to market analysis and general equilibrium theory, although there are many deep connections. Arrow’s legacy as teacher and adviser, of course, goes far beyond his most celebrated students to a virtual legion of economists.

Like Faust, limitless curiosity and passion for knowledge meant that Arrow strove without relenting; but unlike Faust, Arrow needed no redemption. His intellectual integrity was pristine and unparalleled at every stage of his life. His character was as admirable and admired as his intellect. Arrow’s personal and scholarly example continues to inspire, nurture, and challenge.

References

Arrow, K J (1951b), “An Extension of the Basic Theorems of Classical Welfare Economics”, in J Neyman (ed.) Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, Berkeley and Los Angeles: University of California Press.

Arrow, K J (1962a), “Economic Welfare and the Allocation of Resources for Inventions”, in The Rate and Direction of Inventive Activity: Economic and Social Factors edited by R. R. Nelson, Princeton: Princeton University Press.